Question: Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{5}{4}\pi) + i \sin(\frac{5}{4}\pi)] \cdot [7(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{5}{4}\pi) + i \sin(\frac{5}{4}\pi)$ ) has angle $\frac{5}{4}\pi$ and radius $1$ The second number ( $7(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))$ ) has angle $\frac{1}{4}\pi$ and radius $7$ The radius of the result will be $1 \cdot 7$ , which is $7$ The angle of the result is $\frac{5}{4}\pi + \frac{1}{4}\pi = \frac{3}{2}\pi$ The radius of the result is $7$ and the angle of the result is $\frac{3}{2}\pi$.